I know that hash functions make calculations, and the final result is string with fixed length. They are used to verify that files are the same - downloaded files, for example. There is hash presented by the owner of the file, and so the integrity check in this case is comparison of the 2 hashes.
So is this the only file integrity check or there is another thing?
Any integrity check short of a byte-for-byte comparison is unavoidably hash-like. That said, before cryptographic hashes like MD5 became commonly used, simpler algorithms generically called "checksums" were used. Some were standardized (try looking up, for example, CRC32) to one degree or another.
The quality of these methods varies widely. A CRC32 checksum can easily miss corruptions or modifications in a file of any substantial size, for example, while any of the common cryptographic functions (e.g. MD5 or SHA*) are pretty good indicators of integrity.
Be careful about what you use and for what purpose, though -- MD5 and SHA-1 are no longer considered secure against malicious attack (they're just fine if you're just using them as a "better checksum", though).